Let the relative solution error (denoted by RSE) be
\({\sf RSE} = {^{(k)} - ^{\ast } } \big /{^{(0)} - ^{\ast } }\)  at \(^{(k)}\) , where we set \(^{(0)}={0}\) . The experiments are terminated once RSE is less than \(10^{-4}\) , or IT exceeds \(10^{6}\) .
To make the implementations of RGDR and RGDC more efficient, we try to avoid using for-loop structure as far as possible at each iteration. In addition, we execute RGDR and RGDC without explicitly forming the matrices \(\widetilde{A}\)  and \(\widehat{A}\) . Note that RGDR is equivalent to FDBK [1]} when the relaxation parameter \(\theta _1 = 0.5\) . We take the parameters \(\eta _1=0.5\)  in GBK and \(\eta _2=0.1\)  in AMDCD as the same values as that used in [2]} and [3]}, respectively. In RBK and RBCD, the block size is set by \(\widehat{s} = \widetilde{s} = 100\) . To avoid calculating the Moore-Penrose inverse of \(A_{I_k, :}\)  in RBK and GBK and \(A_{:,{\widetilde{\tau }_k}}\)  in RBCD, we use the CGLS algorithm [4]} to solve the least-squares problem. In the following
the symbols RGDR(\(\theta \) ) and RGRK(\(\theta \) ) respectively represent the RGDR and RGRK methods with parameter \(\theta \) . This notation also applies to RGDC and RGRCD. The numerical results for solving the consistent and inconsistent linear systems are reported.
